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Notes -
https://people.seas.harvard.edu/~salil/am106/fall18/A Mathematician%27s Apology - selections.pdf
https://mathoverflow.net/questions/116627/useless-math-that-became-useful
That thread, in general, seems to have a great many examples. Other quotes from it:
I hope this shores up my claim that even branches of maths that their creators (!) or famous contemporary mathematicians called useless have a annoying tendency to end up with practical applications. It's not just in the natural sciences, I've certainly never heard cryptography called a "natural science".
Also, see walruz's claim below , that even what you personally think is useless maths is already paying dividends!
Maths is quite cheap, has enormous positive externalities, and thus deserves investment even if no particular branch can be reliably predicted to be profitable. It just seems to happen nonetheless.
No, I'm sorry, but you really don't know what you're talking about here. The field of pure mathematics is much larger and stranger than you know, and it takes years of intensive study to even reach the frontier, let alone contribute to it. Conic sections and integral transforms are high-school or early university math, and knowing them makes you as much of a pure mathematician as knowing how to change your car's oil filter makes you a CERN engineer. (And, for the record, conic sections were certainly never useless - even other people in that thread you linked called out that ridiculous claim. And non-Euclidean geometry is useful in many other realms than special relativity, like, oh, say, navigating the Earth!)
While there is zero chance of any of the math I linked above being useful, I admit that cryptography isn't the only example of surprising post-hoc utility showing up. As theoretical physics has gotten more abstract (way way beyond relativity), some previously existing high-powered math has become relevant to it. (The Yang-Mills problem, another Millennium Problem, unites some advanced math and physics.) But I absolutely defy the claim that there is a "tendency" for practical applications to show up. Another way to frame the fact that 0.01% of pure math has surprised us by being useful over the last 2,000 years is... that we're right that it's useless 99.99% of the time. I wish I had that much certainty about the other topics we discuss here!
BTW, did you not realize that @walruz was joking? What he linked is a fun Magic: The Gathering construction. If the Twin Primes conjecture is true, then the loop never ends. If it's not true, it does end, after 10^10^10^10^whatever years. It may be slightly optimistic to describe that as "paying dividends"... (Also, the construction only exists because of a card that specifically refers to primes in its rules. You can't claim that math has practical application because it's used to answer trivia questions involving that same math!)
I was leaning into the joke. MTG nerds are a different breed.
On the topic of conic sections, the poster claimed:
They very much didn't start out that way.
That depends on how strict you want to be on the definition of non-Euclidean, spherical geometry, a limited subset, was used in celestial and terrestrial navigation as early as the first century CE, though real usage only boomed in the Age of Sail.
But that was for a very specific purpose, the idea that space itself was non-Euclidean came about much later. That is a lag of about 21 centuries.
I am happy to acknowledge availability and recall bias here. If there are topics in maths that have remained utterly useless and purely theoretical to this day, I am unlikely to have heard of them.
My overall point is that:
Maths is incredibly productive on net.
Even if we do have "99.9%" certainty that a particular field is unlikely to have practical applications, the benefits in the unlikely case that it does are usually substantial. If I came across a normal lottery and saw that my ticket had a 0.1% chance of winning billions, then I'd be spending quite a lot of money on lottery tickets.
Ergo, it is immensely sensible to subsidize or invest in maths as a whole. The expected value from doing so is positive. Our entire society and civilization runs on mathematical advancements.
I have no quibbles with these points! I think what you should take away is that the distribution of potential practicality is far from uniform. There are fields that we can be very, very, very sure aren't practical. If we were horribly utilitarian about things, we could easily, um, "optimize" academic math without losing out on any future scientific progress.
Also, lest my motivations be misunderstood, I'm happy that we fund pure math for its own sake. I took a degree in it. I love it. I just don't want it to be funded under false pretenses.
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I think it's still open to debate whether, in the absence of subsidized pure math research, we'd get the same mathematical advancements "never", "much later", "as soon as we need them", or "practically just as soon".
The fact that everybody thinks of (even their own!) pure math as "useless", right up until it turns out to be the foundation for quantum physics or something, is perhaps the best evidence for "never". I got my PhD in Applied Math (unspoken motto: do you want respect, or do you want job offers?), and it feels almost criminal when you hear about a mathematician coming up with an abstract toy only for someone more focused on science and engineering to come along generations later and say, "whoa, that solves my problem; yoink!"
As evidence for "as soon as we need them": the applied mathematicians haven't been just swiping everything; if you don't find something that solves your problem off-the-shelf, you take what you have and you expand it and tweak it and invent more of it until you do, and in the end you're still proving new theorems, just motivated by "this is how I can guarantee when my new algorithm will converge" rather than "theorems are fun!"
As evidence for "much later": the trouble with "do you want job offers" is that some job offers let you publish more than others, and if you're not getting subsidized via something like academic grants or civilian national lab research, it's downhill from there. Math is in part a cooperative team sport, and it doesn't work as well when you want to score in the "free advancement of human knowledge" basket but you're lucky if you get to shoot for "patent" rather than "trade secret" or "national security" instead.
And as evidence for "practically just as soon", I refer back to "theorems are fun!" There are some people who you can shunt off to a job as a patent clerk and it still won't stop them from playing with tensor calculus; if these are the sort who make the critical-path advances then we still get the advances.
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What about things like quaternions, which suddenly became relevant when we needed to interpolate 3D transforms and do rotations without Gimbal lock? The current best process for calibrating cameras is to use dual quats, which also means needing dual number theory. Were those areas originally expected to be useful for engineering? My understanding is no, but I'm not a mathematician.
That's a good question. I'm not sure of the exact reason quaternions were invented - you can indeed stumble on them just by trying to extend the complex numbers in an abstract way - but the Wikipedia article suggests they were already being used for 3D mechanics within a couple of years of invention. (BTW, "number theory" involves integers, primes, that kind of thing, not quaternions. Complex numbers do show up though.)
You could ask the same question about complex numbers too, but they originally arose from the search for an algorithm to solve cubic equations, which is a fairly practical question. That they later turned out to be essential for electronics and quantum mechanics is a case of some new applications of an already useful math concept.
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